In discussions of climate change, it is often useful to think about the transition of the climate from one state to another, and ask how the magnitude of the response is related to changes in a control parameter (such as the solar constant, or CO2 concentration). This is the classical problem of climate sensitivity, which is intimately connected with assessing the degree to which Earth has the capacity to change. In such an analysis, we typically begin by reducing the “climate” to a single variable, commonly global mean temperature (), and gauge its evolution as a function of the control parameter. Of general interest to society, for example, is how the global mean temperature responds to changes in CO2 concentration. I will build up some theoretical background into climate sensitivity in this article, and move on to observations and modeling results in subsequent posts.
I write the energy balance equation as before, which is typically a close balance between the incoming solar radiation and the outgoing longwave radiation (OLR). The latter depends on the atmospheric temperature profile and opacity, as discussed in previous posts.
where R is the net top-of-atmosphere radiation budget, which is zero in steady state, defined so that R > 0 means the planet cools down, and R < 0, it warms up. We will assume the planet radiates according to where is the surface temperature, and is now introduced to represent the bulk planetary emissivity (dominated by the atmosphere's ability to trap exiting surface radiation…the surface itself has an emissivity very close to one). If the radiation to space emanated exclusively from the ground, then . In reality, the ground radiates about 390 W/m2, and because of the greenhouse effect, about 240 W/m2 exits out of the top of the atmosphere. This establishes a temperature gradient between the surface (~288 K) and the effective emission temperature of the planet many kilometers above the surface (~255 K) corresponding to a bulk emissivity of . We can write,
Suppose we are interested in the case in which we alter the atmospheric greenhouse concentration (say, doubling CO2), by changing , but leave the incoming solar radiation unchanged. Then taking the derivative,
For the present Earth case, we have noted that . The radiative forcing for doubling CO2 is (i.e., every doubling of CO2 reduces the outgoing radiation (or equivalently, the planetary emissivity) by roughly 4 W/m2 at fixed temperature). Solving for the temperature change above, we have:
which reduces to dTs = -(288/4)(-4/240) = 1.2 K, so we expect every doubling of CO2 to change the global temperature by just over a degree.
The analysis above, in the language of climate scientists, is one that ignores feedback effects. In other words, only the temperature adjusts to the change in CO2 concentration (and correspondingly, the outgoing radiation of the planet, which is a pre-requisite to reach a new steady state). The increased or decreased radiation loss to space, as the planet warms or cools (respectively) is an extremely strong stabilizer of climate. We call this the Planck feedback. This is the default reference system used by the climate community by which we can evaluate other feedback effects (and as a reference system, is not often called a “feedback”). Feedbacks do not exert a direct forcing themselves (like external knobs such as CO2) but modify the net TOA energy balance indirectly because of their response to changes in temperature. This can further amplify or dampen the final temperature response. One such feedback already encountered was the ice-albedo feedback. Another is the water vapor feedback, which is likely the most important feedback for modern global warming.
Brief Water Vapor Feedback detour
On Earth, there is a substantial reservoir of water at the surface in the form of ocean or ice. If left undisturbed for a long enough time, water vapor would continue to enter the atmosphere until it reached a point in which its partial pressure equaled the saturation vapor pressure, which is an exponentially increasing function of temperature. Therefore, the content of water vapor in our atmosphere will increase or decrease as temperature goes up or down, respectively; since water vapor is a greenhouse gas, the additional water vapor will amplify whatever caused the initial warming (and same for cooling influences). This idea can be generalized to any greenhouse substance that exists in a temperature regime allowing it to be condense on a planet or moon in consideration (such as methane on Titan). As it happens, the atmosphere does not usually reach saturation, but it turns out that we can think of the fractional change in water vapor (i.e., the slope) as a great approximation to the slope of the saturation vapor curve (i.e., the fixed relative humidity assumption, which is borne out by observation and comprehensive models). This means the actual water vapor content increases quasi-exponentially with temperature, producing a positive (or destabilizing) feedback. A negative (or stabilizing) feedback suppresses the impact of the initial perturbation. I will explore more of the nuances of the water vapor feedback in a later post.
Let ⌀ be a control parameter (say CO2 concentration), and r a feedback factor that depends on temperature, i.e., r = r(T), both of which modify the net TOA energy budget, R. Then,
The new temperature response (now including feedbacks) is therefore:
These expressions may look complicated, but are rather simple when you examine the terms individually. The numerator in equation A is a measure of the radiative forcing (i.e., how the net TOA energy budget is altered with changes in the control parameter, like CO2 concentration). The bracketed term in the denominator is a measure of how the energy budget is altered with changes in temperature, since increase in temperature in almost all circumstances lead to an increase in the outgoing radiation. With no other feedbacks, this slope is .
Climate sensitivity is proportional to the 1/1-f term. As can see in equation B, this is a function of how a feedback variable responds to temperature, and how that alteration actually influences the planetary energy budget. For example, we can replace r with the water vapor concentration of the atmosphere. Water vapor increases with temperature (i.e., is positive) while the outgoing radiation is reduced as water vapor increases ( is negative). This makes the entire expression for f positive, implying a positive feedback.
For f greater than zero, but less than one, the feedback is positive but still smaller than the stabilizing influence of the denominator (the increased radiation loss to space as the planet warms). This is to say that OLR still increases with temperature, even if the atmospheric opacity is increasing with temperature too. This increases climate sensitivity but does not result in a climate system that “runs away.” See the figure below. This diagram shows the outgoing radiation as a function of surface temperature, and two different values for the absorbed incoming radiation. CO2 is fixed in this diagram. You can think of the applied forcing as the change in sunlight between the red and green horizontal lines. The black curve corresponds to a planet that radiates according to (i.e., with only the Planck radiation “feedback” operating). The blue curve corresponds to the same case except with a water vapor feedback, which allows the atmospheric opacity to increase with temperature. This reduces some of the curvature you’d expect with a pure T4 dependence. In steady state, the outgoing radiation curves intersect at the incoming solar radiation lines. Therefore, the climate sensitivity for the black OLR curve case is the difference in the distance between the two blue squares as the solar radiation is increased. The climate sensitivity for the blue OLR curve is the difference in the distance between the red circles as the solar radiation is increased. One can see that you need to increase the temperature more in the latter case to reach the same radiative equilibrium. Put another way, the more sensitive system is less efficient at shedding its infrared emission to space, and so the temperature must increase by that much more in order to come back to balance. One can include shortwave feedbacks into this picture by plotting R on the vertical axis instead of OLR, or adjusting the position of the horizontal lines (as in the ice-albedo post).
Emergent Properties of the Feedback Theory
There are important consequences to the dependence of climate sensitivity on 1/1-f. One is that the effects of feedbacks on climate sensitivity are not simply additive. The second is that how uncertainties in the feedback strengths themselves propagate onto the system response, depends itself on the “true” value of the feedback strength. Some illustrative examples will clear this up.
We can re-write the above expressions as:
where is the “no-feedback” temperature response (i.e., the response to a given forcing with only the reference system operating, such as the Planck response, and no other feedbacks). is called the system multiplier, which amplifies or dampens the sensitivity. If its value is greater than one, feedbacks are net positive; if its value is less than one, feedbacks are net negative. We can now shed light on the two properties outlined above.
1) Suppose two feedbacks are operating, both of which on their own would amplify sensitivity by 20% () relative to the case where only the Planck radiative response was working. This corresponds to both feedbacks factors, f1 and f2, both of which have values of 0.17. The total system response with both feedbacks operating is:
If the feedbacks impacted the system response in an additive fashion, you’d expect a 40% amplification of sensitivity (), but in this case the response is actually 1.52 times the no-feedback reference system. This is because the feedbacks must also act upon each other, and not just on the reference system. The deviation from the additive property would be even more striking if the individual feedback multipliers were larger than 20%.
2) Suppose we don’t know the true value of f for a feedback of interest, but want to know how that uncertainty might correspond to uncertainty in the system response. Then we can differentiate the temperature with respect to a feedback factor,
The dependence on means that for some uncertainty in the feedback, , the uncertainty in the actual temperature response is larger for higher climate sensitivities. This property has been used by Roe and Baker, 2007 (for example) to suggest that narrowing the range of climate sensitivity may be quite difficult (however, see Hannart et al., 2009). A symmetric distribution of the uncertainty in the strength of the feedbacks results in a skewed distribution in the climate sensitivity itself, with a high probability of large values, though it should be noted that there’s no good a priori reason for uncertainties in feedbacks to be symmetric given information from observational and paleoclimate data. As a more practical matter, when using observations, sensitivity distributions depend on the prior assumptions made for the feedbacks. It took the community a while to figure out why some studies showed so dramatically different results from others. It depends largely on whether symmetric uncertainties were assumed for the feedback or the inverse of it. A linear relationship between observational data and the radiative restoring efficiency of the planet, implies that the derivative of the sensitivity with respect to the data becomes zero in the limit of high sensitivity (i.e., it would be harder to distinguish the difference between a 5 and 10 C sensitivity based on looking at the transient response to a volcanic eruption, than it would the difference between a 1.5 and 2 C sensitivity; see e.g., Frame et al., 2005 ).
The f=1 regime
One might expect based on the above equations that as f becomes close to one, the climate sensitivity blows up to infinity and a runaway effect develops (physically, the total feedback effect begins to overwhelm the stabilizing influence of the Planck radiation response). We encountered this with the Snowball Earth discussion, whereby a runaway ice-albedo feedback became sufficiently strong at low enough CO2 concentrations (or sunlight) that it forced the entire planet into an ice-bound state. This runaway feedback then ceases once the planet is covered in ice.
Of course, one cannot use this linear analysis anymore as f becomes too close to one. Feedbacks can be thought of in terms of a Taylor expansion series, and linearization corresponds to just the first term in such an expansion. The next term involve those that depend on df/dT (i.e. the dependence of the feedbacks on the climate state), then d2f/dT2, etc. One can imagine that which feedbacks operate, or the relative importance of those feedbacks, depends on the climate state, and so the linear analysis is no longer valid for large enough perturbations. In the next figure, I plot the net TOA energy budget (this time, R) on the vertical axis vs temperature on the horizontal axis. We apply a radiative forcing, , that brings the climate state from the blue curve to the red curve (for instance, by adding CO2 and lowering the OLR). If there are a set of processes (like an ice-albedo feedback) that happens to give that climate the feedback structure shown in the diagram, then f = 1 actually doesn’t mean a runaway, it means a bifurcation from the first blue circle to the next. f=1 could mean a runaway if there were no equilibrium point on the other side of the bifurcation, but no information can be obtained about where such an equilibrium point might be (or if it exists) without knowledge of the non-linear Taylor expansion terms.
That is a bit of introduction into thinking about feedbacks from a theoretical perspective. Later posts will be sure to elaborate on this, and aim at more practical diagnostics of sensitivity for the modern case, in additions to discussions of other extreme bifurcations aside from Snowball Earth (e.g., the runaway greenhouse).